Meristematic tissues

Lateral meristems occur along the sides of roots and stems, parallel to the long axis, and are responsible for the increase in girth (secondary growth).
Fascicular (intrafascicular) vascular cambium, interfascicular cambium and cork cambium (phellogen) are examples of lateral meristems.
The intercalary meristem occurs between mature tissues, at the bases of internodes or leaves (as in grasses); it increases the length of the organ and regenerates parts removed by grazing herbivores.
Apical and intercalary meristems are primary meristems.
Hence the intercalary meristem is not a lateral meristem.

The jointed stem of grasses such as sugarcane, with solid nodes and distinct internodes, is called a culm.
Intercalary meristem is the meristem that occurs between mature tissues. In grasses it is present at the base of the internodes (just above the nodes).
It is a part of the apical meristem that gets left behind during growth, and it adds to the length of the organ.
Each internode of sugarcane elongates by the activity of its own intercalary meristem.
Since the activity of this meristem differs from one internode to another, the internodes are of different lengths.
The shoot apical meristem is present only at the tip of the stem, and leaf size or the position of axillary buds does not decide the length of an internode.

Every root grows in length at its tip. Behind the root cap lies the root apical meristem, which produces new cells for growth.
This is true whether the root is the radicle (primary root), a tap root, a lateral root or an adventitious root.
So the apical meristem of the root is present in all the roots.

The shoot apical meristem at the tip of the stem forms the terminal bud and produces new leaves and the primary tissues of the stem.
As the leaves form, some cells of the apical meristem are left behind in the axils of the leaves; these give rise to axillary buds, which can grow into branches or flowers.
Lateral meristems (vascular and cork cambium) produce secondary growth in thickness, and intercalary meristems add length at the nodes of grasses.
So axillary and terminal buds are derived from the activity of the apical meristem.